Long memory — where it appears
Named by 17 essays across 10 fields — each of them below, with the objects they name alongside it.
A dependence fitted with the line
Every whitening in this collection reads the dependence off a set of residuals, and residuals are not errors. Fitting the two together recovers most of what that costs, and changes almost nothing about the decision it feeds.
A dependence with a shape
Four ways for errors to repeat, all with the same first lag and nothing else in common. A rule told the errors are a first-order autoregression finds the same number in all four, and is right about one of them.
A family before a fit
A regression's coefficients and one correlation can be maximised together. Replace the correlation with an estimated covariance and there is nothing left for "jointly" to mean — until a set of covariances is named, and the set turns out not to contain the truth.
The fit that takes the memory out
A candidate's residuals report less dependence than its errors do, and how much less is arithmetic rather than noise. The rule used for a good reason reads the series that has lost the most.
The plug-in and the maximum
A tapered covariance estimate sits five and a half log-likelihood units below the maximum of the likelihood it is substituted into. Four fifths of that is what the optimiser would have found if nothing were missing.
How often it matters
The disagreement rate rises by half across the list and the share of disagreements that decide anything falls by nearly the same factor. Their product — how often the tuning list changes which candidate wins — sits at an eighth and does not move.
The window a whitening wants
Every law here is best whitened by a window several times longer than its own memory, including the one whose memory ends at the fourth lag. The three ways of choosing it from the sample all land in the same place, and it is the wrong one.
Three quarters of the way to one search
The pair that started this reads 0.762 on a scale whose one is containment. And the pair that shares nothing but its response reads −0.306, so the sign the earlier field found does not transport at all.
The order the tail is drawn at
A fitted autoregression reproduces the sample exactly at the lags it was fitted on, so everything it says past them is extrapolation — and the order is the dial that decides how much of it there is.
What fitting them together buys
Maximising over the coefficients and the covariance together beats the two-step under one of four dependences and ties under the other three. It is the one the band family contains, and the likelihood said so before any coefficient was compared.
A length read off the whole sequence
The automatic block length reads one number off the sample — the lag-one correlation — and assumes the rest of the dependence is geometric. Reading the whole autocovariance sequence instead removes the assumption and puts the length where the law wants it. At a hundred and twenty rows it buys nothing, even where the assumption is wrong; it costs 3.87 points where the assumption is right, and by 960 rows that premium is 1.15 and the payout under a law the plug-in misreads is 3.77. Under long memory neither rule, nor the best length on each draw, gets within ninety per cent.
A cap that decides nothing
Every band-width list this field priced stopped at thirty lags, and long memory at a large sample is the one dependence whose best band wants to be wider. At 960 rows it does: the draw's own best width lies past the cap on 48.7% of draws. Lift the cap to eighty and the measured charge follows, from 28 lags to 46 — and the error moves by two ten-thousandths, in the wrong direction and inside its standard error, because the surface behind the cap is flat. The field's last conditional fails where it had its best chance.
A ranking that belongs to the interval
Five ways of choosing a block length, read on the coverage of the interval each length builds rather than on the variance it implies. On the percentile interval the variance's verdict mostly survives: under a five-period moving average the plug-in and the whole-sequence rule tie at 120 rows and the sequence rule leads at 480. Build a studentised interval from the same resamples and the lead reverses — the plug-in's longer blocks cover 94.5% against 93.3% — because an interval that widens for its own uncertainty is paid for a long block where a percentile interval is punished for one. Under long memory no length and no interval covers two times in three.
A tail the law gives the pair
Split the over-charge of two searches at their critical value and the interaction — the configurations only the joint search reaches — stays near its average while the overlap climbs. That held under one law. Under a five-period moving average and under long memory it holds for every pair in the order the field reads, to within three units. In the other order, one pair acquires a tail of its own: with the window searched first and the break second, the interaction under long memory climbs from 2.0 at the median to 7.9 at the 99th percentile, four times its average.
A fixed point with no likelihood under it
A tapered band of autocovariances has no likelihood to maximise, so the one rule that can fit its dependence together with the regression is to iterate: read the band off the residuals, refit, read it again. It converges on every draw, in seven to eleven fits at the median, gives back about half the memory the first fit took out of the residuals under a short-memory law, and lowers prediction error by two to four thousandths — five to eight paired standard errors. Under the law a first-order likelihood describes, its maximum still wins; under the two it does not, the band's fixed point beats it.
The surface under every law
A draw-reading charge for a covariance band failed on one law, and the failure was deferred to a law with a steeper error surface. Run under all four laws in the collection, the surface is flat from six lags to thirty under every one, no draw-reading charge tracks anything, and long memory — the law expected to be steepest — is the flattest end to end. What the law moves is the level of the charge, not whether a draw can be read.
The interval the joint fit reports
Maximising a regression's coefficients and an eight-lag covariance band together gives a better estimate than plugging the band in. It also prints an interval that covers 71% to 85% of the time where 95% was promised — on the white predictor too, where every other rule is honest — because it reports almost the precision of the law's own covariance and delivers less than half of that advantage. Counting the eight lags it estimated buys one to two points.
Named alongside it
The objects these essays reach for when they reach for this one.
WhiteningGeneralised least squaresModel selectionDependenceMonte CarloNuisance parameterAutocorrelationCovariance matrixInformation criterionMoving averageRegretTapering